
Sparse Radiance Field (SRF) can be defined as a voxel grid of dimension \(1+d\) , where \(d\)  is the dimension of radiance colors \(\rho _{i,j,k} \in \mathbb {R}^{d} \)  at that specific \((i,j,k) \)  indexed voxel in addition to one dimension for density \(\alpha _{i,j,k} \in \mathbb {R}\) . We assume that the grid is of size \(H\)  in each of the three dimensions: \(\mathcal {X} \in \mathbb {R}^{H^3 \times (1+d)}\) . Since the SRF is sparse, it can be represented with the COO format [1]} as a set of \(M\)  tuples of positive integer coordinates \(\mathbf {c}\in \mathbb {Z}^{+}\)  and features \(\mathbf {f}\in \mathbb {R}^{d+1}\)  with the sparsity of \(1-\frac{M}{H^3}\)  as follows:
\(\begin{aligned}\mathcal {X}_{\text{non-empty}}= \lbrace (\mathbf {c_m},\mathbf {f_m}) \rbrace _{m=1}^{M}\end{aligned}\) 
